Problem: Simplify; express your answer in exponential form. Assume $x\neq 0, a\neq 0$. $\dfrac{{(x^{5}a^{-5})^{2}}}{{(x^{-5}a^{4})^{2}}}$
To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${(x^{5}a^{-5})^{2} = (x^{5})^{2}(a^{-5})^{2}}$ On the left, we have ${x^{5}}$ to the exponent ${2}$ . Now ${5 \times 2 = 10}$ , so ${(x^{5})^{2} = x^{10}}$ Apply the ideas above to simplify the equation. $\dfrac{{(x^{5}a^{-5})^{2}}}{{(x^{-5}a^{4})^{2}}} = \dfrac{{x^{10}a^{-10}}}{{x^{-10}a^{8}}}$ Break up the equation by variable and simplify. $\dfrac{{x^{10}a^{-10}}}{{x^{-10}a^{8}}} = \dfrac{{x^{10}}}{{x^{-10}}} \cdot \dfrac{{a^{-10}}}{{a^{8}}} = x^{{10} - {(-10)}} \cdot a^{{-10} - {8}} = x^{20}a^{-18}$